Introduction to Homotopy Theory

Arkowitz, Martin

doi:10.1007/978-1-4419-7329-0

Cited by 136 publications

(90 citation statements)

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“…which is identical to (4.11) because T • f S (−d) = T by the definition of T . Roughly speaking this means that Λ ∆ℓ can be transformed to Ω ∆ℓ by distorting it in a continuous fashion [22,23]. Below we prove Lemma 4.4 by using the following result.…”

Section: Continuity Of the Return Map In A Cylindrical Topologymentioning

confidence: 97%

The structure of mode-locking regions of piecewise-linear continuous maps: II. Skew sawtooth maps

Simpson

2018

Nonlinearity

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In two-parameter bifurcation diagrams of piecewise-linear continuous maps on R N , mode-locking regions typically have points of zero width known as shrinking points. Near any shrinking point, but outside the associated mode-locking region, a significant proportion of parameter space can be usefully partitioned into a two-dimensional array of nearlyhyperbolic annular sectors. The purpose of this paper is to show that in these sectors the dynamics is well-approximated by a three-parameter family of skew sawtooth circle maps, where the relationship between the skew sawtooth maps and the N -dimensional map is fixed within each sector. The skew sawtooth maps are continuous, degree-one, and piecewise-linear, with two different slopes. They approximate the stable dynamics of the N -dimensional map with an error that goes to zero with the distance from the shrinking point. The results explain the complicated radial pattern of periodic, quasi-periodic, and chaotic dynamics that occurs near shrinking points.

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Section: Continuity Of the Return Map In A Cylindrical Topologymentioning

confidence: 97%

The structure of mode-locking regions of piecewise-linear continuous maps: II. Skew sawtooth maps

Simpson

2018

Nonlinearity

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“…The homotopy maps to zero for each solution state along a defined smooth parameter curve through the (N +1)-dimensional state-space [1,2]. The definition for a linear homotopy is given mathematically in Eq.…”

Section: Continuation To N-body Dynamicsmentioning

confidence: 99%

A Continuation Method for Converting Trajectories from Patched Conics to Full Gravity Models

Bradley

Russell

2014

J of Astronaut Sci

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A method is introduced to transition space trajectories from low fidelity patched conics models to full-ephemeris n-body dynamics. The algorithm incorporates a continuation method that progressively re-converges solution trajectories in systems with incremental changes in the dynamics. Continuation is accomplished through the variation of a control parameter, which is tied to body ephemeris locations and masses, minimum flyby altitudes, and sphere of influence sizes. The intermediate models provide a continuous and differentiable path between solutions in the simplified and n-body dynamics, and the boundary values of the control parameter replicate the patched conics and full-ephemeris models exactly. Each successive step preserves the qualitative properties of the initial guess by successively altering flyby states and body masses. A similar approach to this methodology may be taken with any simplified starting guess, such as a restricted three-body model. Trajectories computed using the patched conics conversion method presented here may include gravity assists and rendezvous with any number of target bodies, so the method is ideal for constructing interplanetary or intermoon tour missions, and is amenable to patching shorter segments together to form longer tours.

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“…We restrict our attention to the case where k is odd and dim(G) = 0 i.e., G is finite. Up to the identification of the group π j (I in(S k /G) ) with π j−1 (F G n−1 (S k \GQ 1 ) × Ω( n−1 1 S k )) described in Theorem 2.5(3), our aim is to (2) there are split short exact sequences:…”

Section: Proofmentioning

confidence: 99%

On the homotopy fibre of the inclusion map $$F_n(X)\hookrightarrow \prod _1^nX$$ F n ( X ) ↪ ∏ 1 n X for some orbit spaces X

Golasiński

Gonçalves

Guaschi

2016

Bol. Soc. Mat. Mex.

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Under certain conditions, we describe the homotopy type of the homotopy fibre of the inclusion map F n (X) ֒−→ n 1 X for the n th configuration space F n (X) of a topological manifold X without boundary such that dim(X) ≥ 3. We then apply our results to the cases where either the universal covering of X is contractible or X is an orbit space S k /G of a tame, free action of a Lie group G on the k-sphere S k . If the group G is finite and k is odd, we give a full description of the long exact sequence in homotopy of the homotopy fibration of the inclusion map F n (S k /G) ֒−→ n 1 S k /G. This paper is dedicated to Professor S. Gitler, for his outstanding work in homotopy theory, his readiness to encourage the scientific progress of colleagues and of institutions, his unwavering intellectual honesty, and his warm friendship.n 1 X x i = x j for all i = j .

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Introduction to Homotopy Theory

Cited by 136 publications

References 0 publications

The structure of mode-locking regions of piecewise-linear continuous maps: II. Skew sawtooth maps

The structure of mode-locking regions of piecewise-linear continuous maps: II. Skew sawtooth maps

A Continuation Method for Converting Trajectories from Patched Conics to Full Gravity Models

On the homotopy fibre of the inclusion map $$F_n(X)\hookrightarrow \prod _1^nX$$ F n ( X ) ↪ ∏ 1 n X for some orbit spaces X

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